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Mirrors > Home > MPE Home > Th. List > chpmatply1 | Structured version Visualization version GIF version |
Description: The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.) |
Ref | Expression |
---|---|
chpmatply1.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
chpmatply1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chpmatply1.b | ⊢ 𝐵 = (Base‘𝐴) |
chpmatply1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chpmatply1.e | ⊢ 𝐸 = (Base‘𝑃) |
Ref | Expression |
---|---|
chpmatply1 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpmatply1.c | . . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
2 | chpmatply1.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | chpmatply1.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | chpmatply1.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2825 | . . 3 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃) | |
6 | eqid 2825 | . . 3 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
7 | eqid 2825 | . . 3 ⊢ (-g‘(𝑁 Mat 𝑃)) = (-g‘(𝑁 Mat 𝑃)) | |
8 | eqid 2825 | . . 3 ⊢ (var1‘𝑅) = (var1‘𝑅) | |
9 | eqid 2825 | . . 3 ⊢ ( ·𝑠 ‘(𝑁 Mat 𝑃)) = ( ·𝑠 ‘(𝑁 Mat 𝑃)) | |
10 | eqid 2825 | . . 3 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅) | |
11 | eqid 2825 | . . 3 ⊢ (1r‘(𝑁 Mat 𝑃)) = (1r‘(𝑁 Mat 𝑃)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | chpmatval 21013 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)))) |
13 | 4 | ply1crng 19935 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
14 | 13 | 3ad2ant2 1168 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
15 | crngring 18919 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
16 | eqid 2825 | . . . . 5 ⊢ (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) | |
17 | 2, 3, 4, 5, 8, 10, 7, 9, 11, 16 | chmatcl 21010 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
18 | 15, 17 | syl3an2 1207 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
19 | eqid 2825 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃)) | |
20 | chpmatply1.e | . . . 4 ⊢ 𝐸 = (Base‘𝑃) | |
21 | 6, 5, 19, 20 | mdetcl 20777 | . . 3 ⊢ ((𝑃 ∈ CRing ∧ (((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) → ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀))) ∈ 𝐸) |
22 | 14, 18, 21 | syl2anc 579 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘(𝑁 Mat 𝑃))(1r‘(𝑁 Mat 𝑃)))(-g‘(𝑁 Mat 𝑃))((𝑁 matToPolyMat 𝑅)‘𝑀))) ∈ 𝐸) |
23 | 12, 22 | eqeltrd 2906 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Fincfn 8228 Basecbs 16229 ·𝑠 cvsca 16316 -gcsg 17785 1rcur 18862 Ringcrg 18908 CRingccrg 18909 var1cv1 19913 Poly1cpl1 19914 Mat cmat 20587 maDet cmdat 20765 matToPolyMat cmat2pmat 20886 CharPlyMat cchpmat 21008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-xor 1638 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-ofr 7163 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-xnn0 11698 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-word 13582 df-lsw 13630 df-concat 13638 df-s1 13663 df-substr 13708 df-pfx 13757 df-splice 13864 df-reverse 13882 df-s2 13976 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-gim 18059 df-cntz 18107 df-oppg 18133 df-symg 18155 df-pmtr 18219 df-psgn 18268 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-rnghom 19078 df-drng 19112 df-subrg 19141 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-ascl 19682 df-psr 19724 df-mvr 19725 df-mpl 19726 df-opsr 19728 df-psr1 19917 df-vr1 19918 df-ply1 19919 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-dsmm 20446 df-frlm 20461 df-mamu 20564 df-mat 20588 df-mdet 20766 df-mat2pmat 20889 df-chpmat 21009 |
This theorem is referenced by: chmaidscmat 21030 cpmidgsum 21050 cpmidgsumm2pm 21051 cpmidpmatlem2 21053 cpmidpmatlem3 21054 chcoeffeqlem 21067 cayhamlem3 21069 cayleyhamilton1 21074 |
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